for nikon coolpix p600--my friend and i both have the same issue of flickering between auto and fi sunset when trying to take pictures. it makes it hard to take an image. it started going more crazy lately. then i switched to manual and thought that worked but some of my images that looked great in the viewfinder big screen area (sorry i dont know the name of it) was later completely overexposed. then i also noticed when i try to view images i only got maybe a tenth of 2 tenths of a second to do so until it turns off. f im really fast i can view maybe 3 (not very good but quick) before it turns off. anyone know what to do?
If you shoot in Manual Mode, then you will have to adjust the settings (shutter speed, aperture, ISO are the main ones) yourself. If you are looking through the viewfinder on your camera - you will see what the lens is seeing. What the sensor (what the camera captures) is seeing might be completely different - so if your settings are wrong, then it will come out overexposed. If you look through the big screen on the back, you'll be seeing what the sensor sees, which is good for checking the settings are correct in manual mode.
As for trying to shoot sunsets, they are pretty hard for a camera to deal with due to the low light levels and changing light. What I can suggest you do is 'tinker' with the manual settings, or shoot in Aperture priority mode (usually marked as AP on the mode dial) and change this number so that the f number is quite high (a number greater than 16 - so something like f/22) - this lets less light in, and should help prevent overexposure. As with all photography, keep practising and changing the settings. Also, look for a white balance setting and adjust it accordingly to match your lighting conditions. This also helps to expose the scene correctly.
As for it shutting off, I'm afraid I've never stumbled across this issue on any other camera - try changing timeout settings in your camera's menu. On my main Nikon camera, I believe it shuts off after a few seconds (but not as quickly as you said) so you can quickly see your pictures before continuing shooting. Are you sure you aren't knocking another button when viewing the pictures? If you accidentally press a 'back' button, or the shutter button, then it will take you back to shooting mode.
im not sure this will work permanently but the contacts might have had some stuff on it in mine. my husband spinned the auto-manual-scene back and forth manually for maybe a minute and it cleared it up, at least for now. you might want to try that and see if it helps.